Problem: Rewrite as a simplified fraction. ${1.63\overline{8} = {?}}$
Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1638.8888...\\ 100x &= 163.8888...\end{align*}}$ Subtracting the two equations, we have: ${900x = 1475}$ ${x = \dfrac{1475}{900}} $ ${x = \dfrac{59}{36}} \text{or} {1 \dfrac{23}{36}}$